Holland (1980) symmetric wind field
The Holland model (1980), widely used
in the literature, is based on the gradient wind balance in mature
tropical cyclones. The wind speed distribution is computed from the
circular air pressure field, which can be derived from the central and
environmental pressure and the radius of maximum winds.
\[
v_r = \sqrt{\frac{b}{\rho} \times \left(\frac{R_m}{r}\right)^b \times
(p_{oci} - p_c) \times e^{-\left(\frac{R_m}{r}\right)^b} + \left(\frac{r
\times f}{2}\right)^2} - \left(\frac{r \times f}{2}\right)
\]
with, \[
b = \frac{\rho \times e \times v_m^2}{p_{oci} - p_c}
\] \[
f = 2 \times 7.29 \times 10^{-5} \sin(\phi)
\]
where,
\(v_r\) is the
tangential wind speed (in \(m.s^{-1}\)),
\(b\) is the shape parameter,
\(\rho\) is the air density set to \(1.15 kg.m^{-3}\),
\(e\) is the base of natural logarithms
(~2.718282),
\(v_m\) the maximum
sustained wind speed (in \(m.s^{-1}\)),
\(p_{oci}\) is the pressure at
outermost closed isobar of the storm (in \(Pa\)),
\(p_c\) is the pressure at the centre of the
storm (in \(Pa\)),
\(r\) is the distance to the eye of the storm
(in \(km\)),
\(R_m\) is the radius of maximum sustained
wind speed (in \(km\)),
\(f\) is the Coriolis force (in \(N.kg^{-1}\)), and
\(\phi\) is the latitude.
Willoughby et al. (2006)
symmetric wind field
The Willoughby et al. (2006)
model is an empirical model fitted to aircraft observations. The model
considers two regions: inside the eye and at external radii, for which
the wind formulations use different exponents to better match
observations. In this model, the wind speed increases as a power
function of the radius inside the eye and decays exponentially outside
the eye after a smooth polynomial transition across the eyewall (see
also Willoughby (1995), Willoughby et
al. (2004)).
\[
\left\{
\begin{aligned}
v_r &= v_m \times \left(\frac{r}{R_m}\right)^{n} \quad if \quad r
< R_m \\
v_r &= v_m \times \left((1-A) \times e^{-\frac{|r-R_m|}{X1}} + A
\times e^{-\frac{|r-R_m|}{X2}}\right) \quad if \quad r \geq R_m \\
\end{aligned}
\right.
\]
with, \[
n = 2.1340 + 0.0077 \times v_m - 0.4522 \times \ln(R_m) - 0.0038 \times
|\phi|
\] \[
X1 = 287.6 - 1.942 \times v_m + 7.799 \times \ln(R_m) + 1.819 \times
|\phi|
\] \[
A = 0.5913 + 0.0029 \times v_m - 0.1361 \times \ln(R_m) - 0.0042 \times
|\phi| \quad and \quad A\ge0
\]
where,
\(v_r\) is the
tangential wind speed (in \(m.s^{-1}\)),
\(v_m\) is the maximum sustained wind speed
(in \(m.s^{-1}\)),
\(r\) is the distance to the eye of the storm
(in \(km\)),
\(R_m\) is the radius of maximum sustained
wind speed (in \(km\)),
\(\phi\) is the latitude of the centre of the
storm, and
\(X2 = 25\).
Adding asymmetry to Holland (1980) and
Willoughby et al. (2006) wind
fields
The asymmetry caused by the translation of the storm can be added as
follows,
\(\vec{V} = \vec{V_c} + C \times
\vec{V_t}\)
where,
\(\vec{V}\) is the
combined, asymmetric wind field,
\(\vec{V_c}\) is symmetric wind field,
\(\vec{V_t}\) is the translation speed
of the storm, and
\(C\) is
function of \(r\), the distance to the
eye of the storm (in \(km\)).
Two formulations of C proposed by Miyazaki et al. (1962) and Chen
(1994) are implemented.
Miyazaki et al. (1962)
\(C = e^{(-\frac{r}{500} \times
\pi)}\)
Chen (1994)
\(C = \frac{3 \times R_m^{\frac{3}{2}}
\times r^{\frac{3}{2}}}{R_m^3 + r^3 +R_m^{\frac{3}{2}} \times
r^{\frac{3}{2}}}\)
where,
\(R_m\) is the radius
of maximum sustained wind speed (in \(km\)).
Boose et al. (2004)
asymmetric model
The Boose et al. (2004) model,
or “HURRECON” model, is a modification of the Holland (1980) model (see also Boose et
al. (2001)). In addition to adding
asymmetry, this model treats of water and land differently, using
different surface friction coefficient for each.
Wind speed
Wind speed is computed as follows,
\[
v_r = F\left(v_m - S \times (1 - \sin(T)) \times \frac{v_h}{2} \right)
\times \sqrt{\left(\frac{R_m}{r}\right)^b \times e^{1 -
\left(\frac{R_m}{r}\right)^b}}
\]
with, \[
b = \frac{\rho \times e \times v_m^2}{p_{oci} - p_c}
\]
where,
\(v_r\) is the
tangential wind speed (in \(m.s^{-1}\)),
\(F\) is a scaling parameter for friction
(\(1.0\) in water, \(0.8\) in land),
\(v_m\) is the maximum sustained wind speed
(in \(m.s^{-1}\)),
\(S\) is a scaling parameter for asymmetry
(usually set to \(1\)),
\(T\) is the oriented angle
(clockwise/counter clockwise in Northern/Southern Hemisphere) between
the forward trajectory of the storm and a radial line from the eye of
the storm to point \(r\),
\(v_h\) is the storm velocity (in \(m.s^{-1}\)),
\(R_m\) is the radius of maximum sustained
wind speed (in \(km\)),
\(r\) is the distance to the eye of the storm
(in \(km\)),
\(b\) is the shape parameter,
\(\rho = 1.15\) is the air density (in \(kg.m^{-3}\)),
\(p_{oci}\) is the pressure at outermost
closed isobar of the storm (in \(Pa\)),
and
\(p_c\) is the pressure at
the centre of the storm (\(pressure\)
in \(Pa\)).
Wind direction
Wind direction is computed as follows,
\[
\left\{
\begin{aligned}
D = A_z - 90 - I \quad if \quad \phi > 0 \quad(Northern \quad
Hemispher) \\
D = A_z - 90 - I \quad if \quad \phi \leq 0 \quad(Southern \quad
Hemispher) \\
\end{aligned}
\right.
\] where,
\(D\) is the
direction of the radial wind,
\(A_z\) is the azimuth from point r to the
eye of the storm,
\(I\) is the
cross isobar inflow angle (\(20\) in
water, \(40\) in land), and
\(\phi\) is the latitude.
Wind fields comparison
Here, we compare wind fields generated by different models that can
be used in StormR for the same time and location (tropical
cyclone Pam near Vanuatu)
sds <- defStormsDataset()
## Warning in checkInputsdefStormsDataset(filename, sep, fields, basin, seasons, : No basin argument specified. StormR will work as expected
## but cannot use basin filtering for speed-up when collecting data
## === Loading data ===
## Open database... /private/var/folders/v1/6y176c8d2jz06qr8rd3jq_480000gn/T/RtmpSAC0zx/Rinst11d381788fbb5/StormR/extdata/test_dataset.nc opened
## Collecting data ...
## === DONE ===
st <- defStormsList(sds = sds, loi = c(168.33, -17.73), names = "PAM", verbose = 0)
PAM <- getObs(st, name = "PAM")
pf <- spatialBehaviour(st, product = "Profiles", method = "Holland", asymmetry = "None", verbose = 0)
terra::plot(pf$PAM_Speed_41, main = "Holland (1980)", cex.main = 0.8, range = c(0, 90))
terra::plot(countriesHigh, add = TRUE)
lines(PAM$lon, PAM$lat, lty = 3)
pf <- spatialBehaviour(st, product = "Profiles", method = "Willoughby", asymmetry = "None", verbose = 0)
terra::plot(pf$PAM_Speed_41, main = "Willoughby et al. (2006)", cex.main = 0.8, range = c(0, 90))
terra::plot(countriesHigh, add = TRUE)
lines(PAM$lon, PAM$lat, lty = 3)
pf <- spatialBehaviour(st, product = "Profiles", method = "Holland", asymmetry = "Miyazaki", verbose = 0)
terra::plot(pf$PAM_Speed_41, main = "Holland (1980) + Miyazaki et al. (1962)", cex.main = 0.8, range = c(0, 90))
terra::plot(countriesHigh, add = TRUE)
lines(PAM$lon, PAM$lat, lty = 3)
pf <- spatialBehaviour(st, product = "Profiles", method = "Willoughby", asymmetry = "Miyazaki", verbose = 0)
terra::plot(pf$PAM_Speed_41, main = "Willoughby et al. (2006) + Miyazaki et al. (1962)", cex.main = 0.8, range = c(0, 90))
terra::plot(countriesHigh, add = TRUE)
lines(PAM$lon, PAM$lat, lty = 3)
pf <- spatialBehaviour(st, product = "Profiles", method = "Holland", asymmetry = "Chen", verbose = 0)
terra::plot(pf$PAM_Speed_41, main = "Holland (1980) + Chen (1994)", cex.main = 0.8, range = c(0, 90))
terra::plot(countriesHigh, add = TRUE)
lines(PAM$lon, PAM$lat, lty = 3)
pf <- spatialBehaviour(st, product = "Profiles", method = "Willoughby", asymmetry = "Chen", verbose = 0)
terra::plot(pf$PAM_Speed_41, main = "Willoughby et al. (2006) + Chen (1994)", cex.main = 0.8, range = c(0, 90))
terra::plot(countriesHigh, add = TRUE)
lines(PAM$lon, PAM$lat, lty = 3)
pf <- spatialBehaviour(st, product = "Profiles", method = "Boose", verbose = 0)
terra::plot(pf$PAM_Speed_41, main = "Boose et al. (2004)", cex.main = 0.8, range = c(0, 90))
terra::plot(countriesHigh, add = TRUE)
lines(PAM$lon, PAM$lat, lty = 3)
